Inauguraldissertation zur
Erlangung der W¨urde eines Doktors der Philosophie vorgelegt der
Philosophisch-Naturwissenschaftlichen Fakult¨at der Universit¨at Basel
von
Ian James Rouse aus Birmingham, United Kingdom
Basel, 2018
Originaldokument gespeichert auf dem Dokumentenserver der Universit¨at Basel
edoc.unibas.ch
Prof. Dr. Stefan Willitsch und Prof. Dr. Philipp Treutlein
Basel, den 13.11.18
Prof. Dr. Martin Spiess Dekan
Experiments involving trapped ultracold matter are of great interest to a diverse range of fields, from spectroscopy and quantum computing to ultra- cold chemistry. Hybrid traps allowing for the simultaneous confinement of charged and uncharged matter extend the scope of these experiments, but have not yet benefited from the miniaturisation of the trapping architectures demonstrated for traps which only confine either ions or neutral particles.
This miniaturisation greatly enhances the spatial resolution of the forces with which the trapped particles are manipulated, and this thesis details the design and fabrication of a prototype miniaturised hybrid trap to take advantage of this increased precision. The co-trapping of ions and neutral particles leads to multiple mechanisms by which the energy distributions of the trapped ions may deviate from thermal statistics, which have previously been treated largely empirically. In this thesis, these effects are explored numerically and analytically to provide a theoretical framework for this behaviour through the formalism of superstatistics. The results derived here explain the deviations from thermal statistics observed in precision spectroscopy experiments and resolve outstanding questions about both the mechanism by which ions ac- quire a non-thermal energy distribution during buffer gas cooling with neutral atoms and the analytical form of this distribution. This significantly improves the ability to correctly interpret the results of experiments, and is applicable not only to the hybrid chip trap developed here, but to hybrid ion-neutral traps in general.
I would like to thank my supervisors, Professor Stefan Willitsch and Profes- sor Philipp Treutlein, for the opportunity to study here in Basel and their guidance in the work presented here. The past and present members of the Willitsch group provided great company both in and out of university. With- out them the time here would have been much less enjoyable, and much less would have been achieved. The patience and hard work of the staff of the mechanical and electronic workshops — Grischa Martin, Philipp Kn¨opfel and Georg Holderied — in translating my crude sketches and vague requests into reality was greatly appreciated, as was the construction of lasers by Dr Ana- toly Johnson and Dr Andreas J¨ockel.
The construction of the surface-electrode ion trap took multiple rounds of design and trials of different techniques, and I am very grateful to Dr Nadeem Rizvi of Laser Micromachining Ltd., Armin Stumpp and Beat L¨uscher of the Fachhochschule Nordwestschweiz, and Dr Yves P´etremand of the Swiss Cen- ter for Electronics and Microtechnology for all their assistance and patience during this process. I must also thank Dr Cezar Harabula for his assistance with wirebonding and Dr Roman Schmied for his input during the chip design.
Funding from the Swiss Nanoscience Institute for this project (P1214) is gratefully acknowledged, as is funding from the Swiss National Science Foun- dation as part of the National Centre of Competence in Research, Quantum Science, and Technology (NCCR-QSIT) and Grant No. 200020 175533.
Last but certainly not least, thanks to my friends and family for supporting me throughout my time in Basel and reminding me that there is life outside of university.
1 Introduction 1
1.1 Cooling and trapping particles . . . 1
1.2 Applications of cold trapped particles . . . 5
1.3 Outline of the thesis . . . 8
2 Theory of particle cooling and trapping 10 2.1 Laser cooling . . . 11
2.1.1 Interaction of light with a two-level atom . . . 11
2.1.2 The scattering force . . . 16
2.1.3 Heating effects . . . 19
2.2 Particle trapping . . . 22
2.3 The magneto-optical trap . . . 24
2.4 Magnetic traps . . . 26
2.5 Ion traps . . . 30
2.5.1 Radiofrequency ion traps . . . 30
2.5.2 The Mathieu equation . . . 34
2.6 Particle interactions . . . 48
2.6.1 Ion-ion interactions . . . 52
2.6.2 Collective effects of uncharged particles . . . 54
2.6.3 Interactions in hybrid traps . . . 55
2.7 Tsallis statistics and superstatistics . . . 55
3 A hybrid ion-atom trap on a chip 62 3.1 Background . . . 62
3.2 Design of the chip . . . 63
3.2.3 Atom chip . . . 69
3.3 Construction . . . 73
3.3.1 Ion chip . . . 73
3.3.2 Atom chip . . . 74
3.3.3 Heatsink and U-bar . . . 75
3.3.4 Assembling the hybrid chip . . . 77
3.3.5 Vacuum system . . . 78
3.3.6 Optical setup . . . 81
3.3.7 Electronics . . . 83
3.3.8 Experimental control . . . 83
3.4 Conclusions . . . 84
3.5 Appendix: Technical drawings . . . 84
4 Numerical simulations of trapped particles 88 4.1 Motivation . . . 88
4.2 Implementation of molecular dynamics simulations . . . 89
4.2.1 Trapping potentials . . . 89
4.2.2 Laser cooling . . . 90
4.2.3 Particle interactions . . . 96
4.2.4 Background gas . . . 98
4.2.5 Trap imperfections . . . 98
4.3 Simulations of ions in the hybrid trap . . . 99
4.4 Simulations of atoms in the hybrid trap . . . 102
4.4.1 On-chip MOT . . . 102
4.4.2 Magnetic conveyor sequence . . . 106
4.5 Conclusions . . . 109
4.A Numerical approximation to the secular motion . . . 109
5.2 Force model . . . 112
5.2.1 Collisions with background gas . . . 113
5.3 Temperature fluctuations . . . 117
5.3.1 Superstatistical velocity distributions . . . 120
5.3.2 Consequences for studies of cold chemistry . . . 124
5.3.3 Influence of micromotion . . . 125
5.4 Summary and conclusions . . . 126
5.A Averaging collisions over impact parameters . . . 127
5.B Time-averaged velocity distribution . . . 128
5.C Time-averaged Arrhenius rate constant . . . 131
6 Statistical mechanics of a single ion in a neutral buffer gas 132 6.1 Energy change during ion-neutral collisions . . . 134
6.1.1 Motion of an ion in a radiofrequency trap . . . 134
6.1.2 Ion-neutral collisions . . . 135
6.2 Energy distributions . . . 142
6.2.1 Reduction to one dimension . . . 142
6.2.2 The multiplicative model . . . 146
6.2.3 Requirement for a lower bound . . . 153
6.3 Tsallis statistics . . . 157
6.3.1 Parameter estimation . . . 162
6.3.2 Estimation of the Tsallis exponent in the presence of EMM . . . 165
6.4 Localised buffer gases . . . 173
6.4.1 Collisions at the centre of the trap . . . 173
6.4.2 Localised buffer gases . . . 176
6.4.3 The energy distribution due to localised buffer gases . . 180
6.B Numerical methods . . . 190
6.C Averaging overη . . . 191
6.D Moments of superstatistical distributions . . . 196
6.E The Bessel-Tsallis distribution . . . 197
6.F Mathematica notebooks . . . 200
7 Conclusions and outlook 214
Bibliography 217
A List of publications 228
2.1 Rabi oscillations and the rate-equation model . . . 13
2.2 Energy level diagram of a three level atom . . . 16
2.3 Velocity-dependent excited state population and the scattering force . . . 17
2.4 Magneto-optical trap field and laser polarisation . . . 25
2.5 Quadrupole magnetic field from a single wire . . . 28
2.6 Magnetic fields from U and Z wire configurations . . . 29
2.7 Field and laser configuration for a mirror-MOT . . . 30
2.8 Geometry and pseudopotential for a linear Paul trap . . . 33
2.9 Geometry and pseudopotential for a surface electrode ion trap . 33 2.10 Phase-space trajectory of an ion in a Paul trap . . . 40
2.11 Trajectory and Fourier spectrum of an ion including excess mi- cromotion . . . 44
2.12 Impact parameter and collision trajectories . . . 50
2.13 Tsallis and thermal distributions . . . 60
3.1 Schematic of the ion chip . . . 65
3.2 Electric potential of the RF electrodes of the ion chip . . . 67
3.3 Pseudopotential plots of the ion chip . . . 69
3.4 Schematic of the atom chip . . . 70
3.5 Axial potentials generated by the atom chip . . . 72
3.6 Initial prototypes and final design of the ion chip . . . 74
3.9 The assembled hybrid chip . . . 77
3.10 Vapour cell assembly . . . 78
3.11 Schematic of the vacuum system . . . 79
3.12 Transitions used for trapping of rubidium and optical setup around the hybrid chip . . . 82
3.13 Final design of the ion chip . . . 85
3.14 Final design of the atom chip . . . 86
3.15 Design of the fused silica cuvette . . . 87
4.1 Upper-state population and temperature evolution in the state- tracking laser cooling model . . . 95
4.2 Cooling rate and equilibrium temperatures in the state-tracking and friction models . . . 95
4.3 Secular temperature following a collision of an ion with back- ground hydrogen. . . 100
4.4 Simulated CCD image of 200 ions in the hybrid trap . . . 101
4.5 Phase-space distributions for a U-wire MOT . . . 106
4.6 Phase space acceptance of the on-chip mirror-MOT . . . 106
4.7 Potentials generated by the magnetic conveyor belt . . . 108
4.8 Position and temperature of atoms during the magnetic con- veyor belt sequence . . . 108
5.1 Recooling of a Coulomb crystal to equilibrium after a coolision 115 5.2 Comparison of experimental to simulated CCD images . . . 116
5.3 Increase in secular temperature due to collisions with back- ground gas . . . 119
5.4 Recooling dynamics of an ejected ion into a crystal . . . 119
5.5 Non-thermal velocity distribution obtained as a result of a col- lision . . . 121
6.1 Components of the mean secular energy for an ion interacting with a buffer gas, with and without excess micromotion (EMM) 141 6.2 Distributions of the energy-transfer factorηin ion-atom collisions145 6.3 Energy distribution of an ion interacting with a zero-temperature
buffer gas . . . 152 6.4 Energy distributions obtained with buffer gas temperatures of
0K and 1fK . . . 156 6.5 Invariance of the scaled energy distributions due to changing
either the buffer gas temperature or the magnitude of excess micromotion with the other set to zero . . . 156 6.6 Comparison of the predicted and observed energy distributions 164 6.7 Comparison of the values ofnT andhβi found numerically to
that predicted using the multiplicative model. . . 164 6.8 Energy distributions with and without excess micromotion . . . 168 6.9 The Tsallis exponent in the presence and absence of excess
micromotion . . . 169 6.10 Effect of non-thermal buffer gas velocity distributions on the
secular energy distribution . . . 170 6.11 The Tsallis exponent in the presence of both EMM and a ther-
mal buffer gas . . . 172 6.12 Energy change due to ion-neutral collisions at the centre of a
radiofrequency trap. . . 176 6.13 Effects of a localised buffer gas on the secular phase at the time
of a collision . . . 179 6.14 Energy distribution for a calcium ion interacting with a rubid-
ium buffer gas in the hybrid chip . . . 184 6.15 Mean and mean-square values ofηfound numerically and com-
pared to the analytical model . . . 194
as a function of the mass ratio . . . 195
Introduction
1.1 Cooling and trapping particles
Everyday matter consists of a vast number of particles. A litre of water, for example, contains on the order of 3×1025 molecules constantly collid- ing with each other and undergoing random thermal motion. Under these circumstances, it is essentially impossible to investigate the behaviour of a single particle, and instead this must be inferred from the properties of the entire ensemble. The situation would be greatly improved if, instead of the unmanageably large number of particles present under typical conditions, the size of the collection of atoms or molecules could be limited to a much smaller number, or even a single particle. At room temperature, a free particle moves with a typical velocity of hundreds of meters per second, plus or minus a few more hundred. To be able to accurately measure the properties of the particle, it is therefore necessary to both slow the particle down and to re- duce this wide spread in the range of velocities. That is, the particle must be cooled to as low a temperature as possible. Since the particle cannot be stopped entirely, it is necessary to ensure that it remains in a narrow region
of space. A collision with the walls of a container would lead to the particle thermalising with the container and regaining the energy removed by cooling, and thus a method of remotely applying a confining force to the particle is required. Thus, we conclude that to be able to precisely study a particle, it is necessary for the particle to be both cooled and trapped.
The first issue that must be overcome is cooling the particle to a sufficiently low temperature. One route to do so is through the supersonic expansion, in which the expansion of particles forced through a nozzle into vacuum results in an overall cooling, but a high forward velocity [1]. The resulting packet of particles must then be decelerated through some other means, e.g, through the application of time-dependent electric or magnetic fields [2–4]. Alterna- tively, the technique of laser cooling can be used to cool particles if they have a suitable transition between quantum states which can be excited by laser light. In this method, a force is generated due to the transfer of momentum during the absorption and re-emission of photons by the particle [5, 6]. The rate at which the particle absorbs photons, and thus the force experienced by the particle, is modulated by the Doppler effect, and by tuning the frequency of the laser such that the particle absorbs more photons when it is travel- ling towards the laser than when it is travelling away a net cooling force is produced. The force exerted on the atom due to the absorption of a single photon is minute, and so to achieve a measurable result requires the scattering of a large number of photons. For this process to work efficiently, the particle must return to its original state after emitting a photon with a probability of close to unity, i.e., it must have a closed optical cycle. If it does not, then the particle may become trapped in a metastable state, requiring the use of ad- ditional lasers to close the optical cycle and reducing the efficiency of cooling [6]. Consequently, laser cooling has so far been demonstrated only for cer- tain atoms and molecules, typically, atoms with a single valence electron or molecules with favourable vibrational structure [6–8]. However, when it can be employed, laser cooling enables the cooling of particles to temperatures
of less than a millikelvin. Moreover, in contrast to the supersonic expansion, laser cooling is a continuous process and so ensures that the sample of par- ticles remains cold for an extended period of time, and does not require the use of a decelerator to produce a sample which is both cold and slow.
We next move on to the issue of preventing the escape of the particle.
There are a wide variety of possible routes to do so. A tightly-focused laser detuned from the resonance of a transition can be used to generate a confining potential through the optical dipole force [6, 9]. The scattering of photons from a laser, meanwhile, cannot directly trap particles, but when combined with an inhomogenous magnetic field the result is the magneto-optical trap, which simultaneously cools and traps particles. Neutral atoms which are sufficiently cold and are in a suitable hyperfine quantum state can be di- rectly confined in a purely magnetic trap [10]. If the particle has a non-zero charge, then it is possible to apply strong forces to the particle using electric fields, however, a static electric field alone is not sufficient to provide three- dimensional confinement as a consequence of Maxwell’s equations. There are two widely-used solutions to this issue. By combining a static, quadrupolar electric potential with a magnetic field the result is the Penning trap [11].
Stable trapping is also possible using an electric field oscillating at frequen- cies on the order of megahertz, as first demonstrated in the Paul trap and later generalised to a family of radiofrequency ion traps [11, 12]. These ra- diofrequency ion traps, along with the magneto-optical and magnetic traps, are discussed in more depth in Chapter 2.
As a general rule, the magnetic field generated by a current-carrying wire or the electric field from an electrode held at a certain potential decays as a power of the distance from the element generating it, with the character- istic length scale of this decay proportional to the size of the device. At greater distances, the forces generated become weaker. Moreover, at these large distances, the fields from neighbouring wires or electrodes are no longer well-resolved, and it is no longer possible to precisely manipulate the trapped
particles by applying different currents or voltages to these elements. Con- versely, if the particle is too close to the device, then the force from one particular element is much greater than from the rest, and again the ability to precisely control the particle is lost. There is therefore great interest in miniaturising the devices, as this enables fine control at a much closer dis- tances which, in turn, allows for greater forces to be applied to the trapped particles. This has lead to the development of miniaturised forms of the magnetic traps used for neutral atoms and the radiofrequency traps used for charged particles, referred to as atom chips and ion chips respectively [13, 14].
These planar devices trap particles at distances of less than a millimeter from the surface of the trap, and as such are capable of applying large, precisely shaped forces to the ions or neutrals.
The trapping techniques mentioned above typically work only for a narrow class of particles. Magneto-optical traps rely on the presence of a closed optical cycle at a wavelength which can be addresed using lasers, and only recently has this technique been extended from atoms to some molecules [15, 16]. Magnetic traps require atoms to be prepared in a particular hyperfine state, and the Penning and Paul traps both capture only charged particles. In order to study the interactions between different types of particles it is therefore necessary to combine these different mechanisms into hybrid traps. This can be achieved by constructing a device which produces two different traps simultaneously.
In many cases, these hybrid traps are a combination of a radiofrequency ion trap with some mechanism for trapping neutral particles, as demonstrated by numerous groups [17–21]. As both the ion trap and neutral magnetic traps can be miniaturised into chip-based architectures to increase the precision with which they manipulate trapped particles, it seems a natural extension to combine these together to produce a hybrid chip trap. Such a device would be a useful tool for ion-neutral experiments with a higher degree of control than is achievable in the macroscopic hybrid traps, and this thesis presents the design and construction of a prototype of a hybird ion-neutral chip trap.
1.2 Applications of cold trapped particles
Having established that it is indeed possible to trap particles at low temper- atures, we now address the issue of why this is desirable. A single atom in vacuum offers a system for which the quantum-mechanical behaviour can be calculated to a very high level of accuracy, and an experimental realisation of this system offers the ability to confirm these calculations. Not only does this allow for the validation of the underlying theories and the investigation of any potential modifications to the standard model, e.g. fifth forces, but it also enables the measurement of physical constants such as the electron-to-nucleon mass ratio and the fine structure constant and the detection of any changes in these values [22, 23]. The isolation from the environment is invaluable for studies of antimatter which would otherwise rapidly annihilate [24, 25], and for ordinary matter this isolation enables quantum states to be sufficiently long-lived that they may be used as a building block for quantum computers [26, 27].
The next logical step is to introduce a second particle, which need not be of the same type as the first. Since both particles are confined to the same region of space, they will eventually reach the same location and undergo a collision. At high energies, the collisions between particles can usually be modelled as that of two hard spheres bouncing off each other like billiard balls, but at the low energies achievable under these circumstances this is no longer the case [28]. The interaction potential between the two particles can be mapped to a cross-sectional area describing the effective size of one particle from the perspective of the other, and so determining how likely it is for a collision to occur. Depending on the relative velocity between the two particles, this cross section may increase or decrease, resulting in a rate of collisions which is highly dependent on the collision energy and the underlying potential describing the interactions between the two particles. If a collision does occur, then a variety of outcomes are possible, ranging from a simple transfer of energy to a change in the state of the particles, all of which
have their own cross section. It therefore may be the case that collisions at a particular range of energies mostly lead to a transfer of energy, wheras at other energies a reaction between the two particles is more likely. The possibility for a reaction to occur at very low temperatures gives rise to the field of ultracold chemistry [29]. Consider, for example, a collision between a charged particle A+and an uncharged particleB. This may lead to either a simple exchange of momentum, or it may result inAcapturing an electron fromB. This charge exchange reactionA++B →A+B+ is one of the most elementary chemical reactions, and in an ion-neutral hybrid trap can be studied under precisely controlled conditions [30, 31]. The cross section is not measured directly in an experiment, and it is instead the rate of reactions which is measured. This quantity is averaged over the entire range of collision energies present in the system, and therefore the resolution is improved by decreasing this range as far as possible, i.e., working with cold particles. The theoretical rate constant is calculated by averaging the cross section over the distribution of collision energies, and compared to the experimentally measured value to assess the accuracy of the theory [28]. More complex reactions can also be investigated under conditions which would not be feasible in a traditional beaker filled with chemicals, enabling, for example, the study of the reaction rates of different spin isomers of water [32], the formation of exotic metal oxides [33], and the conformer-dependent reactions of isolated organic compounds [34]. It is also possible to construct molecules directly from their component atoms via photoassociation or Feshbach resonance [35, 36], offering another route to the production of ultracold molecules of interest to spectroscopy and quantum computing, amongst other applications [37].
There is no reason why this should stop at only two particles. For large ensembles of particles at the low energies considered here, states of mat- ter occur which would not otherwise be observable. An ensemble of bosons at a sufficiently high phase-space density may undergo a transition to form a Bose-Einstein condensate and exhibit collective quantum behaviour [38].
When multiple ions are simultaneously confined, the resulting one-component plasma exhibits a variety of phases depending on the ratio of the average ki- netic energy to the average distance between particles, and at sufficiently low temperatures an ordered crystalline phase known as a Coulomb crystal is formed, taking a variety of shapes from a string of ions to a series of con- centric ellipsoidal shells depending on the number of trapped ions and the anisotropy of the trapping potential [39–44]. The ions are localised in discrete lattice sites and so may be individually addressed using tightly-focused lasers, and the rate at which ions are lost from the crystal used to measure rates of chemical reactions [45]. Co-trapping different species allows for the study of reactions as described above, but also enables the sympathetic cooling of particles which cannot otherwise be cooled. This may either be achieved in hybrid traps for, e.g., the cooling of ions by neutral atoms, or a single trap if this can be designed to confine two different masses simultaneously. This latter approach has been demonstrated for radiofrequency traps containing multiple species of ion, with a light laser-cooled ion used as a coolant for heavy ions lacking a closed optical cycle [46], and the sympathetic cooling of ions by atoms has also been achieved [19, 21, 47], but is limited in the range of masses which can be successfully cooled [48]. Hybrid traps in which the atoms have formed a Bose-Einstein condensate have also been the focus of some studies, with experiments investigating the rate of collisions and theory predicting the formation of mesoscopic structures of shells of increased atomic density around the ion [49–51].
As a result of the small particle numbers involved in trapping experiments, the conditions which typically lead to thermal equilibrium, i.e., a large num- ber of weakly interacting particles, no longer hold. Consequently, it cannot be assumed that the familiar Maxwell-Boltzmann distribution (or, for very cold ensembles, the Fermi-Dirac or Bose-Einstein distributions) describes the velocity of trapped particles, and attempting to calculate reaction rates based on this assumption will lead to inaccurate values [52]. Likewise, any property
which depends to some extent on the velocity or the energy of the trapped par- ticles will, when measured experimentally, reflect an average over the range of velocities or energies present in the system. It is therefore critical that when these distributions do not follow the standard thermal statistics, the ac- tual forms for these distributions are known to ensure that the experimental results can be accurately compared to theoretical values. Ions held in radiofre- quency traps, in particular, have been observed to show marked deviations from thermal statistics under a range of circumstances [19, 52–60]. Rare heat- ing events significantly increase the energy of the trapped ions, which takes a finite amount of time to be removed by laser cooling. This leads to cycles of heating and recooling, with the result that the ions exhibit a non-thermal velocity distribution on experimental timescales [52, 57]. The time-dependent potential used for radiofrequency traps leads to a heating mechanism when a trapped ion collides with co-trapped neutral atoms, and the ion may gain energy even if the neutral atom is effectively at rest. This has been ob- served to produce a power-law energy distribution for the ion, and establishes an upper limit on the mass of the neutral buffer gas which can be used to cool the ion before a runaway heating mechanism leads to loss of the ion [19, 47, 48, 53, 54, 56]. Recently, it has been demonstrated that this runaway heating can be prevented by using a buffer gas with a non-uniform density by ensuring that collisions occur preferably at the centre of the trap [20, 21, 55].
It is therefore of interest to derive both how this occurs, and the resulting steady-state energy distribution of the ion to better interpret experimental results.
1.3 Outline of the thesis
This thesis has two complementary goals: the design of a miniaturised hybrid chip trap and the study of the dynamics of trapped ions under conditions other than thermal equilibrium. The former allows for the increased control
over ion-neutral systems by enabling more precise manipulation of the trapped particles, while the latter provides a more accurate interpretation of the result of experiments performed on trapped ions by finding the distribution of ener- gies present. Taken together, this allows for the design of experiments with a high degree of control over the collision energies and an accurate knowledge of the spread of the values of these energies, greatly improving the ability to compare theory to experiment. Chapter 2 provides an overview of how the laser cooling and the trapping of both charged and uncharged particles to produce a hybrid system is performed, and gives a brief introduction to the formalism of superstatistics used to describe the statistical mechanics of non- equilibrium systems. In Chapter 3, the details of the design and fabrication of a miniaturised hybrid ion-atom chip trap is presented, and the molecular- dynamics simulations used to characterise the capability of chip to trap both charged and uncharged particles are presented in Chapter 4. The remaining two chapters, 5 and 6, discuss two different cases in which trapped ions may exhibit non-thermal energy distributions. In Chapter 5, these result from rare heating events due to the collisions of trapped particles with background gas of a high temperature, resulting in a cycle of heating and recooling that causes a deviation from thermal statistics on experimental timescales. Chap- ter 6 details the sympathetic cooling of an ion by an ultracold buffer gas and shows how micromotion interruption leads to a non-thermal distribution for the secular energy of the ion, with the form of this distribution analytically derived based on the change in energy during a collision for both a uniform buffer gas and a buffer gas held in a harmonic potential, such as that gener- ated by the hybrid chip trap. Finally, the results of this thesis are summarised in Chapter 7 and it is briefly discussed how the hybrid chip trap can be used to investigate the range of distributions derived in Chapter 6.
Theory of particle cooling and trapping
A number of different techniques must be employed simultaneously to produce an ensemble of ultracold particles, and the goal of this chapter is to introduce the theory behind these techniques. A semi-classical model of Doppler laser cooling is presented in Section 2.1 to establish the mechanism by which the particles may be cooled down through the application of off-resonant laser light. The means through which these ultracold particles may then be trapped in one location is discussed in Section 2.2, providing an overview of magneto- optical, magnetic, and radiofrequency electric traps. The forces through which trapped particles interact with each other and the effects of these interactions are discussed in Section 2.6. The chapter concludes with an introduction to the formalism of superstatistics used to describe the properties of systems with fluctuating temperatures.
2.1 Laser cooling
In most cases, if a laser is focused on an object, then the energy absorbed by that object is dissipated as heat, and the temperature of the object rises.
Counterintuitively, under the correct circumstances a laser may be used to cool atoms down to temperatures a fraction of a degree above absolute zero [6, 61]. This effect relies on a combination of three properties: that atoms interact with light only at specific frequencies, that the frequency of light observed by an atom may be shifted by the Doppler effect, and that when an atom scatters a photon a small quantity of momentum is transferred from a fixed direction to a random direction. In the following sections, an overview of how these properties are combined to result in cooling of an atom is presented.
2.1.1 Interaction of light with a two-level atom
The scattering of light by atoms is an inherently quantum process and is linked to the transition of the system between discrete energy states. Consider a system with only two available energy levels: a ground state|1iand an excited state|2i, separated by an energyE=~ω0. In the absence of external forces, an atom prepared in one of the two states will remain there. A time-dependent potential, however, may couple these two states together, such that the atom’s state is given by a superposition of these two states,|Ψi=c1|1i+c2|2i. To be specific, we will assume that this time-dependent potential is the electric field of monochromatic light of frequencyωl, and that the two relevant states are the ground and first excited electronic state of an atom. The mathematical treatment of how the system evolves due to the light is non-trivial, and so here a summary of the main results from Ref. [62] are presented. The density matrix can be defined fromc1, c2 as [62],
p11 p12
p21 p22
!
= c∗1c1 c1c∗2 c∗1c2 c∗2c2
!
, (2.1)
wherec∗ is the complex conjugate ofc. The diagonal elements of this matrix are the populations, that is, the probability for the atom to be in that state, while the off-diagonal elements are the coherences between the states. These are then expressed in terms of the components of the Bloch vector [62],
u v w
=
p12e−iδt+p21eiδt
−i(p12e−iδt−p21eiδt) p11−p22
(2.2)
whereδ=ωl−ω0 is the detuning of the angular frequency of the light from the atomic resonance. The advantage of this form is that the time-evolution ofu, v, wis given by a set of ordinary linear differential equations,
d/dt
u(t) v(t) w(t)
=
δv(t)
−δu(t) + ΩRw(t)
−ΩRv(t)
, (2.3)
where ΩR is the Rabi frequency defined in terms of the laser intensityI, the saturation intensity of the transition Isat, and the natural linewidth of the transition Γ12 as,
Ω2R= 1 2
I Isat
Γ212. (2.4)
The saturation intensity is given by [62], Isat=2π2
3
~c
λ3Γ12, (2.5)
wherec is the speed of light andλ0= 2πc/ω0 is the wavelength of the tran- sition.
Solving the differential equations given by Eq. (2.3) with the initial con- ditionsu=v = 0, w= 1, corresponding to an atom initially in the ground
0.0 0.2 0.4 0.6 0.8 1.0 0.0
0.2 0.4 0.6 0.8 1.0
t(us) p22(t)
Figure 2.1: The population of the upper state due to transitions from the lower state caused by an applied electric field. Three cases are shown: the Rabi oscillations due to an applied field in the absence of noise (blue solid line), the result of adding spontaneous emission to the Rabi oscillations leading to damped oscillations (red dashed line), and a simplified rate-equation model (black dotted line). In all cases, the Rabi frequency is taken to be 5×2πMHz, the detuning is given by 1×2πMHz, and where appropriate, the lifetime of the upper state is 1×2πMHz.
state, and using the normalisation conditionp11+p22= 1 results in, p22= Ω2R
Ω2R+δ2sin2
pΩ2R+δ2
2 t
(2.6) from which it can be seen that the atom undergoes coherent oscillations, known as Rabi oscillations, between the ground and excited state, as depicted in Fig. 2.1. In practice, the atom does not continue to oscillate forever due to the presence of noise, which results in transitions from the excited state back to the ground state. This process of spontaneous emission occurs at a rate equal to Γ12. The set of equations given by Eq. (2.3) may be modified
to include this decay, producing the Optical Bloch Equations (OBE) [62],
d/dt
u(t) v(t) w(t)
=
δv(t)−Γ212u(t)
−δu(t) + ΩRw(t)−Γ212v(t)
−ΩRv(t)−Γ12(w(t)−1)
. (2.7)
The analytical solution for p22(t) is plotted in Fig. 2.1 to demonstrate the results of damping. As a consequence of the decay of the upper state, the initial oscillations rapidly die out, and the system tends towards a steady- state with the probability to be in the upper state given by,
p22(t→ ∞) = Ω2R/4
δ2+ Ω2R/2 + Γ212/4. (2.8) For understanding the mechanism of laser cooling, it is useful to view the atom as existing either in the upper state or the lower state, and undergoing discrete transitions between the two. In this model, three types of transition are possible, corresponding to the absorption or emission of a photon. In the ground state, the atom may absorb a photon and enter the excited state.
Two types of emission from the upper state are possible, corresponding to the fact that the loss of population from the upper state can be due to either the coherent driving or random decay back to the ground state, refered to as stimulated and spontaneous emission respectively. In this discrete model, it can be useful to employ the Einstein rate equations to describe the evolution ofp22 [62],
˙
p22=γa(1−p22)−γep22−Γ12p22, (2.9) where γa, γe are the rate constants for absorption and stimulated emission respectively, and are equal for the two-level system considered here, γa = γe=γ, while the rate of spontaneous emission is given by Γ12. By requiring that the steady-state value ofp22 is equal to that predicted from the OBE,
the value forγ is found to be,
γ= Γ12Ω2R
4δ2+ Γ212. (2.10)
The solution to Eq. (2.9) is plotted in Fig. 2.1. By construction, it results in the same steady-state population, but the initial oscillations are missing and the initial rise in population occurs at a greater rate than it does in the solutions to the OBE. Nonetheless, these differences are only important for a very short period of time, and the rate equation model is particularly useful for numerical simulations of laser cooling, see Chapter 4.
Before proceeding further, a few complications should first be addressed.
Any given atom has far more than two electronic states, and if the atom has multiple valence electrons it is very difficult to find a simple optical cycle as required here. For the most part, laser cooling is limited to either atoms of alkali metals, or the singly-charged ions of alkaline earth metals, although some exceptions exist for other atoms and some simple molecules. Even for the hydrogen-like atoms, it is fairly common for there to exist an energy level which is intermediate in energy between the two states used for laser cooling and which has the same parity as the ground state, see Fig. 2.2 for a schematic of the level structure. Consequently, transitions between this state and the ground state are forbidden, but the atom may spontaneously decay from the excited state into this state, where it becomes trapped. This may be overcome by introducing a laser to repump the atom from|3iback to|2i, ensuring that the laser cooling may continue. Moreover, if the atom has a non-zero total angular momentumF in any of the electronic states, then this electronic level is split into 2F + 1 Zeeman sublevels representing the different projections of this angular momentum onto a quantisation axis. The energies of these states are shifted in applied magnetic fields and by the electric field of the light itself, altering the scattering rate.
|1i
|2, mF = 1i
|2, mF = 0i
|2, mF =−1i
|3i
Figure 2.2: A schematic of the energy levels of a three level atom, showing the two levels used for lasing cooling and a third level which the excited state may decay into via spontaneous emission. The excited state is shown as being split into three Zeeman states, which are degenerate in the absence of a magnetic field but have been shown offset from each other for clarity. The curved arrows indicate possible routes for spontaneous emissions.
2.1.2 The scattering force
So far, the position and velocity of the atom have been neglected. As long as the atom has a reasonably large energy, these may be taken to be classical, continuous variables with well-defined values, which for simplicity are assumed to not vary during the absorption and emission of a single photon [62]. In the frame of reference of the atom, the frequency of the laser is altered due to the Doppler effect, with the result that the detuning from resonance depends on the velocity of the atom and is given byδ(v) =ωl−ω0−k·v, where v is the velocity of the atom andkis the wavevector of the laser. If the frequency of the laser is less than the frequency of the transition (red-detuned), then the light is brought back into resonance when the atom is moving towards the source of the light. Thus, the atom has a greater probability to be in the excited state when it is moving towards the laser than it does when the atom is moving away, see Fig. 2.3(a).
Each photon carries a small quantity of linear momentum, given by ~k, which must be conserved before and after both absorption and emission.
-100 -50 0 50 100 0.0
0.1 0.2 0.3 0.4 0.5
v[m/s]
p22
(a)
-100 -50 0 50 100
-4×107 -2×107 0 2×107 4×107
v[m/s]
Fs/(ℏk)[s]
(b)
Figure 2.3: (a) The steady-state population of the excited state as a function of the velocity of the atom due to driving by an off-resonant laser. (b) The velocity- dependent force acting on a particle due to a pair of counterpropagating lasers, both of which are red-detuned from a transition at 397 nm. In both cases, the laser intensity is equal to the saturation intensity, the linewidth of the transition is taken to be Γ12= 20.7×2πMhz, and the detuning from resonance isδ= Γ12.
Hence, the momentum of the atom must also change by this amount whenever a photon is absorbed or emitted. When a photon is absorbed, this requires that the atom’s momentum is increased in the direction in which the pho- ton was travelling. The reverse is true for stimulated emission, since in this case conservation of momentum requires that the atom is accelerated in the opposite direction to the k-vector of the emitted photon. Thus, absorption followed by stimulated emission leads to no overall effect. Spontaneous emis- sion, however, occurs in a random direction. By symmetry, the momentum change due to the spontaneous emission averages to zero, and so absorption followed by spontaneous emission leads to an overall change in the atom’s momentum. Note that the momentum due to the emitted photon cannot be entirely neglected, as it establishes the minimum momentum of the atom at the recoil limit [62], and also leads to heating of the atom, see Section 2.1.3.
To simplify the description of the cooling, we initially neglect these effects.
The rate at which photons are scattered by the atom is given by the rate of spontaneous emission from the excited state multiplied by the probability
for the atom to be in the excited state. Thus, the scattering rate is given by,
Rs(v) = Γ12p22, (2.11)
wherep22is the population of the upper state taking into account the velocity- dependent detuning. Each scattering event causes the momentum of the atom to change by~k, and so the rate of change of the momentum, i.e. the scat- tering force, is given by,
Fs=madv
dt =~kRs(v), (2.12)
wherema is the mass of the atom. Substituting in the steady-state value for p22 produces [62],
Fs=~k1 2
I Isat
4(ωl−ωΓ02−k·v)2 12
+II
sat + 1
. (2.13)
The force acting on the atom due to absorption of photons is in the direction of travel of the laser, repelling the atom from the laser source. If the laser is red-detuned from resonance, then this force is largest when the atom is moving towards the laser, as this is when the Doppler shift cancels out the detuning.
Conversely, if the atom is moving away from the laser, the force is significantly weaker, but some photons will still be scattered. As such, an atom initially moving towards the laser will be decelerated until it is brought to a standstill, then accelerated in the direction of the laser until it is sufficiently far from resonance that there is no longer any force acting on the atom. This can be prevented by employing a second, counter-propagating laser. Approximating that scattering from the two beams occurs independently of each other, the net force is given by adding together the scattering force from each beam, producing the result shown in Fig. 2.3(b). If both lasers are red-detuned by the same amount and of equal intensity, then the scattering will occur
preferentially from the beam opposing the motion of the atom, and so the net force decelerates the atom regardless of the direction in which it is moving.
To simplify the analysis of the cooling, we take a one-dimensional model in which the velocity and the wave-vector are parallel, and use scalar quantities vandk. Aroundv= 0, the scattering force from the sum of the two beams is approximately a linear function ofv, and so a first-order Taylor expansion can be used to gain more insight into the cooling process. Eq. (2.13) is expanded for each laser with detuningδi and intensity Ii,
Fs,i≈ci−λiv, (2.14)
where,
ci=~ki
Γ12
2
I/Isat
4δ2/Γ2+ 1 +I/Isat
(2.15) and,
λi= 4~ki2δ/Γ12
I/Isat
(4δ2/Γ2+ 1 +I/Isat)2. (2.16) The total scattering force is then given byFs=madv
dt ≈c−λv, withλ=P
iλi
and likewise forc. It follows that the velocity of the atom evolves according to,
v(t) =c/λ+ [v(0)−c/λ]e−λt/ma, (2.17) i.e., an exponential decay with a rate of λ/ma to an equilibrium value of c/λ. This would suggest that, if the two beams are of the same intensity and detuning such that c = 0, the velocity of the atoms is reduced to zero. In reality, this is prevented by the stochastic nature of the scattering force which ensures that the atom has some probability of having a non-zero velocity.
2.1.3 Heating effects
Two heating processes must be taken into account. The first of these is the recoil of the atom as a result of spontaneous emission. Every time a photon
is scattered, the velocityvalong a given axis increases by,
∆v= ~k
ma(cosφsinθ,sinφsinθ,cosθ)T, (2.18) whereθ, φare random variables describing the orientation of the direction of the emitted photon, andk =|k| is the scalar magnitude of the wavevector.
Since the emission is isotropic, the mean increase in the velocity is zero, but the variance of each component of the velocity grows as a function of time.
When the scattering is from a single laser, this is given by, hvj(t)2i − hvj(t)i2=1
3(~k/ma)2Rs(v)t (2.19) where the factor of 13 reflects the fact that the spontaneous emission occurs isotropically and so the momentum increase is shared equally between the x, y, zaxes [62]. The second effect is less immediately obvious and is a con- sequence of the fact that the absorption of photons is a random process [62], and applies only to the direction parallel to the k-vector of the laser, which we take to be thezaxis. Consider an atom interacting with a single laser for a period of time t in which Rs(vz) remains constant, and the atom absorbs Np photons. Excluding spontaneous emission, the change in the velocity of the atom is given byNp(0,0,~k/ma). Since a random number of photons is scattered during this time,Np is not a fixed number but instead is a random variable, and thus, vz(t) is also a random variable. Taking, for simplicity, vz(0) = 0, the variance ofvz(t) is given by,
hvz(t)2i − hvz(t)i2= (~k/ma)2[hNp2i − hNpi2]. (2.20) If each scattering event is independent of the others and the rate remains unchanged, thenNp follows Poisson statistics with a mean value ofhNpi = Rs(vz)t. The mean-square value for Poisson statistics is given by hNpi2 =
hNpi2+hNpi[63], and so,
hvz(t)2i − hvz(t)i2= (~k/ma)2Rs(vz)t. (2.21) Thus, the width of the velocity distribution, and hence the temperature, in- creases due to the random number of photons scattered per unit time.
As a result of these stochastic terms, it is no longer possible to write down a simple expression for the velocity as a function of time. Instead, a statistical approach is used to calculate the probability distribution of vj, denoted∗ fvj(vj), representing the probability for vj to fall in the interval [vj, vj+dvj] [63]. During a period of time which is long compared to 1/Γ12, such that many photons are scattered, but short compared to 1/λsuch that the velocity does not change significantly, the sum of the random changes invj
may be approximated by a normal distribution by the central limit theorem.
The evolution of the velocity of the atom may then be described in terms of a Langevin equation [6],
dvj
dt =− λ ma
vj+σξ(t), (2.22)
where ξ(t) is white noise with a Gaussian distribution and zero mean, σpa- rameterises the strength of this noise, and it has been assumed that there are two counter-propagating beams of equal intensity and detuning to eliminate the radiation pressure term. This is an example of the Ornstein-Uhlenbeck process, and the steady-state probability distribution forv is given by [64],
fvj(vj)∝e−
λv2 j
maσ2, (2.23)
∗This is an abuse of notation. Conventionally, a random variableX is distinguished from a possible value of this variablex, and the distribution is denotedfX(X =x). The shorthand is employed here to avoid either needing to find an unambiguous pair of upper and lowercase letters for each random variable or introducing additional subscripts, superscripts, or diacritical marks to distinguish between a random variable and a realisation of that variable.
i.e. vj is distributed according to a Gaussian distribution. This is the distri- bution expected for a particle in thermal equilibrium with a heat bath at a temperatureT = m2kaσ2
Bλ, wherekBis Boltzmann’s constant, leading to the con- vention of referring to the width of the velocity distribution as a temperature.
Three-dimensional cooling may be achieved through the optical molasses con- figuration, in which there is a pair of counterpropagating beams along each axis. Since these may well have different detunings or intensities, it is not necessarily the case that the same temperature is achieved for the motion along each axis, and the analysis is complicated by the fact that scattering from one beam necessarily reduces the rate of scattering from the others due to saturation of the transition. In the special case where all six lasers have identical detunings and intensities, and saturation effects can be ignored, the minimum steady-state temperature is given by [62],
TD= ~Γ12
2kB
, (2.24)
referred to as the Doppler temperature. The actual temperature achievable may be much smaller than this due to the additional cooling mechanisms available from the interaction between multiple lasers and the fact that the atom is not a simple two level system [65]. If the atom has Zeeman structure in both the ground and excited states, then sub-Doppler cooling mechanisms are possible due to light-induced shift in the energies of these states [62], and exploiting these processes allows a reduction of the energy of the atom to close to the limit set by the recoil of a single photon [62].
2.2 Particle trapping
As discussed in Section 2.1.3, the velocity of a laser-cooled particle does not reach zero, but fluctuates around this value due to the stochastic nature of photon scattering. Thus, the particle diffuses away from its original position,
analogous to the Brownian motion of a particle in solution. After a suffi- ciently long period of time, it would leave the region of space in which it can interact with the lasers and become lost. This may be prevented by applying a position-dependent force to the atom to ensure that it remains trapped.
A simple example of a suitable force is one which is a linear function of the distance from the centre of the trap, Fx=−mω2x, and which leads to har- monic motion of the trapped particles at a frequencyω. As long asω <<ΩR, then the velocity of the atom can be treated as essentially fixed during each scattering event (the weak-binding approximation), and the theory discussed in the previous section is unchanged [66]. The evolution of the position and velocity of the atom is given by a pair of coupled Langevin equations [64],
dx
dt =v, (2.25)
and,
dv
dt =−mω2x+ c m− λ
mv+σξ(t). (2.26) The steady-state probability distributions forxand vare given by Gaussian distributions [64], and so the position and velocity can both be described using thermal distributions. Any residual radiation pressure displaces the equilibrium position of the atom, but the mean velocity remains zero. Thus, if an external trapping potential is used, it is possible to cool the trapped atoms using only one laser per axis, or indeed a single laser if this has a non-zero component along each of the principle axes of the trap and the frequencies of these axes are non-degenerate [5, 67]. The question therefore arises as to how a suitable trapping potential can be generated. A wide range of different methods are possible, of which the three relevant to this thesis will be discussed: the magneto-optical trap, the magnetic trap, and the radiofrequency ion trap.
2.3 The magneto-optical trap
An elegant method to produce a trapping potential is by making the scattering rate depend on the position of the particle as well as the velocity, so that the lasers provide both cooling and a restoring force. This may be done by exploiting the change in the energy levels of the atom when exposed to a magnetic field through the Zeeman effect, resulting in the magneto-optical trap (MOT). For an atom with a total angular momentumF, there are 2F+1 Zeeman sublevels labelled withmF =−F,−F+ 1, ..., F −1, F, wheremf is the projection of the angular momentum onto the quantisation axis. In a weak magnetic field, which is assumed to define the quantisation axis, the potential energies of these states are given by [62],
E(mF) =gfµBmf|B(r)| (2.27) where|B(r)|is the magnitude of the magnetic field at the location of the atom r,µB is the Bohr magneton, andgf is a proportionality constant. As a result of the fact that each of these sublevels has a different energy, the laser light is detuned from resonance by a different amount for each sublevel. Furthermore, if the magnetic field varies as a function of the position of the atom, then the scattering rate for each of the Zeeman states (except formf = 0) is position dependent. A suitable choice of field is the quadrupole field obtained from a pair of coils in the anti-Helmholtz configuration, see Fig. 2.4(a), which produces a field of the form,
B=Bg(1 2x,1
2y,−z), (2.28)
whereBg is the gradient and is usually on the order of 0.1 T/m [62].
To demonstrate how a field of this form alters the scattering rate, consider an atom at rest with ground stateF = 0, excited state F0 = 1, interacting with a laser of k-vector (0,0,−k). In this case, the scattering rate from each
B
x z
B B
E
F, mf = -1 F, mF=1
(a) (b) E
Figure 2.4: (a) A pair of current loops in the anti-Helmholtz configuration, pro- ducing a quadrupolar magnetic fieldB. (b) A schematic of the effects of circularly polarised light on the handedness of the precession of the angular momentum vector Faround a quantisation axis defined by the local magnetic fieldB. In the lab frame, the direction of rotation ofF matches that of the light used to drive the transition, but the handedness of the rotation aroundB, and hence the Zeeman shift, depends on the orientation ofB.
mf state is given by, Rs(mf, z) =Γ12
2
I/Isat
1 +I/Isat+ 4(ωl−ω0−gfµBmfBg|z|)2/Γ212. (2.29) As|z|increases then the Zeeman shift leads to a change in the overall detuning from resonance, and so alters the scattering rate. If unpolarised light is used, the same scattering force is obtained at both z and −z for a given final state mf, that is, Fs(mf, z) = Fs(mf,−z). This symmetry is broken by the selection rules for the transition if circularly polarised light is used, which cause only transitions to a certain value ofmf to be allowed. If the k-vector of the light points in the same direction as the quantisation axis and the electric field of the light rotates in a clockwise direction relative to its own k-vector, then transitions to the mf = 1 state are driven, and those to the mf = −1 state are suppressed [62]. At the opposite point in the trap, the quantisation axis is reversed, and so the same laser instead drives transitions to themf =−1 state, see Fig. 2.4(b). Consequently, although themf = 1 state may be in resonance with the laser at a pair of points±z, this transition can
only be driven at one of the two. Conversely, the transition to themf =−1 state is allowed at the opposite location, but as this has a different detuning from resonance the scattering rate is different. Thus, the scattering force is no longer symmetric with respect to inversion of z, and can be made to be weakly anti-trapping forz <0, and restoring for z >0. This situation is reversed for a laser with the oppositek-vector, and so the sum of the scattering force from the pair of lasers is overall restoring for allz. The combination of three pairs of counter-propagating lasers, as used in molasses cooling, and an anti-Helmholtz coil pair is sufficient to generate three-dimensional cooling and confinement, although the model of scattering presented above breaks down due to the interaction of the multiple lasers. Typically, the cooling is efficient enough that a MOT may capture atoms with velocities up to vc ≈70 m/s [62], enabling their loading from atomic vapour.
2.4 Magnetic traps
The random scattering of photons used to generate the magneto-optical trap also leads to a continuous rate of heating, thus limiting the achievable tem- perature. It would therefore be beneficial if, once the atoms have been cooled down to a sufficiently low temperature, they may then be trapped without requiring further scattering of photons and then potentially cooled to even lower temperatures. This may be done by again exploiting the Zeeman shift due to an inhomogenous magnetic field, as this produces a position-dependent energy. If the atoms are pumped into a state withmf >0 then trapping is possible around a point at which the field magnitude is at a minimum. Take, for example, the quadrupole field given by Eq. (2.28). For this field,|B| has a value of 0 atx=y =z = 0 and increases linearly around this point. This would suggest that the quadrupole field can be used for purely magnetic trap- ping. There is, however, a significant disadvantage to the use of a quadrupole field. Close to the centre of the trap, themf states are very close in energy,
and so transitions may occur due to field fluctuations which cause transitions from mf states which can be trapped to a state which cannot [38]. Thus, it is necessary to produce a field which has a non-zero minima. This may be achieved through use of the Ioffe-Pritchard trap which has a non-zero mini- mum [38, 62]. The disadvantage of this trap is that it requires large currents (≈100 A) to operate, as a result of the macroscopic scale of the trap.
The magnetic field gradient at a distance r generated by a wire carrying a currentI decays as a function of I/r2, and so achieving a significant force requires either a very high current or for the atom to be very close to the wire. If the atoms are sufficiently close then even a very low current of a few amperes may be sufficient to generate a magnetic trap. In this regime, three-dimensional trapping may be achieved through the use of a planar set of wires – an atom chip – in place of the macroscopic coils required for traditional magnetic trapping [68].
To illustrate how these traps works, first consider the magnetic field in the xz plane by a wire carrying current in the +y direction. A homogenous bias field can be used to cancel out this field at a particular point, and around this point the field is approximately a quadrupole field, see Fig. 2.5. By itself, this does not offer confinement along they axis, as the magnetic field has no component in this direction. It is therefore necessary to add an additional two wires carrying current in thexdirection in order to provide axial confinement [68]. Two configurations for these wires are widely used. The first is the U-wire, shown in Fig. 2.6(a), in which one wire carries current in the +x direction and the other in the−xdirection. In the plane equidistant between these two wires, the component of the magnetic field in theydirection is equal to zero, as the magnetic field from the two wires cancels at this point. The second configuration is the Z-wire (Fig. 2.6(b)), for which the two fields do not cancel and so provide axial confinement with a non-zero field minimum.
Thus, the U-wire serves as an approximation to the quadrupole trap, while the Z-wire is equivalent to the Ioffe-Pritchard trap. These basic configurations
-100 0 100 0
100 200
300 -100 0 100
0 100 200 300
x/μm
z/μm
(a)
-100 0 100
0 100 200
300 -100 0 100
0 100 200 300
x/μm
z/μm
(b)
Figure 2.5: (a) The magnetic field in thexz plane created by a current passing through a wire parallel to theyaxis, where the wire is located atx= 0, z=−200µm and has a current of 4A. (b) The field shown in (a) combined with a homogenous bias field to produce an approximate quadrupolar field. The magnitude of this bias field chosen such that atx= 0, z= 150µm the total magnetic field is zero.
may then be combined and extended to produce a flexible variety of trapping potentials, see Chapter 3 for more details.
The depth of these traps is typically only on the order of a few millikelvin, and so loading cannot be performed directly from background vapour. It is therefore necessary to cool the atoms before they are trapped. The opera- tion of a standard MOT is hindered by the fact that, unless a transparant substrate is used, the chip itself blocks the light which would be used to cool in the direction perpendicular to the surface of the chip. Fortunately, it is possible to achieve a three-dimensional MOT by reflecting two of the beams from the surface of the chip, see Fig. 2.7(a). The handedness of the circular polarisation of the beam is reversed by the reflection, and if the lasers are correctly aligned with a magnetic quadrupole field then this ensures that the total scattering force allows for confinement of particles [68]. In theory, the required quadrupole field may be generated by an on-chip U-wire in combi- nation with a bias field. However, this is effectively only in a small region of space compared to the large quadrupole fields achieved through a pair of
external coils, limiting the efficiency with which atoms may be collected from background vapour. Consequently, either external coils or a specially designed U-wire must be used for the first MOT stage [69]. Once a sufficient number of atoms are collected they may be transferred into an on-chip MOT, which has the advantage of higher field gradients and more precise control of the location of the trapped atoms to optimise loading into a purely magnetic trap.
I
Bbias
I
Bbias (a) U wire
(b) Z wire
x y
x y
Figure 2.6: Schematics of the U (a) and Z (b) wires used for the generation of magnetic traps for neutral particles, and the magnetic field profile along theyaxis generated by the wires parallel to thexaxis.
